Question: Zach scores a ringer $40\%$ of the time that he throws a horseshoe. Let $R$ be the number of throws it takes Zach to score his first ringer in a game. Assume the results of each throw are independent. Find the probability that it takes Zach $2$ or fewer throws to score his first ringer. You may round your answer to the nearest hundredth. $P(R\leq 2)=$
Solution: Without a fancy calculator On each throw: $P({\text{ringer}})=0.4$ $P(\text{no ringer}})=0.6$ If it takes Zach $2$ or fewer throws to score his first ringer, here are the possible sequences of throws: ringer no ringer, ringer We can find the probability of each sequence and add those probabilities together. $\begin{aligned} P({\text{ringer}}) &= {0.4}\\\\\\ P(\text{no ringer}},{\text{ringer}}) &= (0.6})({0.4})\\\\&=0.24\\\\\\ P(R\leq2) &= 0.4+0.24\\\\&=0.64 \end{aligned}$ [Is there another way?] $P(R\leq 2) = 0.64$